# 1991 AJHSME Problems/Problem 14

## Problem

Several students are competing in a series of three races. A student earns $5$ points for winning a race, $3$ points for finishing second and $1$ point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?

$\text{(A)}\ 9 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11 \qquad \text{(D)}\ 13 \qquad \text{(E)}\ 15$

## Solution

There are two ways for a student to get $11$: $5+5+1$ and $5+3+3$. Clearly if someone gets one of these combinations someone else could get the other, so we are not guaranteed the most points with $11$.

There is only one way to get $13$ points: $5+5+3$. In this case, the largest score another person could get is $5+3+3=11$, so having $13$ points guarantees having more points than any other person $\rightarrow \boxed{\text{D}}$.

## Solution 2

If someone gets $11$ points, the possible combinations are $5,5,1$ and $5,3,3$ If he gets $5,3,3$ then someone else can be $5,3,3$ which would not guarantee victory. If we have 13 points, the only way to make this is $5,5,3$. There is no way to get any number of points higher than this, so the answer is $\boxed{\text{D}}$.---stjwyl